If \(y = \frac{1+x}{1-x}\), find \(\frac{dy}{dx}\).

If \(y = \frac{1+x}{1-x}\), find \(\frac{dy}{dx}\).

Answer Details

To find \(\frac{dy}{dx}\) when \(y = \frac{1+x}{1-x}\), we need to use the quotient rule of differentiation, which states that if \(y = \frac{u}{v}\), then $$ \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}. $$ In this case, we have $$ u = 1+x \qquad \text{and} \qquad v = 1-x. $$ Taking the derivatives of these functions, we get $$ \frac{du}{dx} = 1 \qquad \text{and} \qquad \frac{dv}{dx} = -1. $$ Substituting into the quotient rule formula, we get \begin{align*} \frac{dy}{dx} &= \frac{(1-x)(1) - (1+x)(-1)}{(1-x)^2} \\ &= \frac{1-x+1+x}{(1-x)^2} \\ &= \frac{2}{(1-x)^2}. \end{align*} Therefore, the correct answer is \(\frac{2}{(1-x)^2}\). Option (A) is the correct answer.